OFFSET
0,1
COMMENTS
This is the first of the three angles (in radians) of a unique triangle that is right angled and where the angles are in a harmonic progression: Pi/(2+2*sqrt(2)) (this sequence), Pi/(2+sqrt(2)) (A193373), Pi/2 (A019669). The angles (in degrees) are approximately 37.279, 52.721, 90. The common difference between the denominators of the harmonic progression is sqrt(2).
LINKS
FORMULA
Equals Pi/(2+2*sqrt(2)).
Equals Integral_{x=0..Pi/2} cos(x)^2/(1 + sin(x)^2) dx = Integral_{x=0..Pi/2} sin(x)^2/(1 + cos(x)^2) dx. - Amiram Eldar, Aug 16 2020
Equals 4*Sum_{k >= 0} (-1)^k/((4*k + 1)*(4*k + 2)*(4*k + 3)). - Peter Bala, Jul 15 2024
EXAMPLE
0.6506451422...
MAPLE
evalf(Pi/(2+2*sqrt(2)), 120); # Muniru A Asiru, Sep 30 2018
MATHEMATICA
N[Pi/(2 + 2*Sqrt[2]), 100]
Realdigits[Pi/(2 + 2*Sqrt[2]), 10, 100][[1]] (* G. C. Greubel, Sep 29 2018 *)
PROG
(PARI) default(realprecision, 100); Pi/(2+2*sqrt(2))
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)/(2 + 2*Sqrt(2)); // G. C. Greubel, Sep 29 2018
CROSSREFS
KEYWORD
AUTHOR
Frank M Jackson, Jul 24 2011
STATUS
approved